Exploring Bosonic Mediator of Interation at BESIII

We present a comprehensive investigation on the possibility of the search for new force mediator $X$ boson in $e^+e^-$ collision and $J/\psi$ decay at the BESIII experiment. The typical interactions of $X$ boson to leptons and quarks are explored. And, the exclusion limits on the reduced coupling strength parameters as functions of $X$ boson mass are presented. Assuming the decay width $\Gamma_X$ of $X$ boson lies in $0.1\sim10$ MeV, we find that the exclusion limits on the reduced coupling strengths fall in the region of $10^{-3} \sim 10^{-2}$, depending on $m_X$, for various hypotheses in the literature. In our estimation, events of $Z^0$-like $X$ boson produced in $e^+e^-\to X\gamma$ may reach about $(10^{9}\sim10^{10}) \times \epsilon^2$ per year, which is accessible in nowadays BESIII experiment.


I. INTRODUCTION
The strong and electroweak interactions in between ordinary matter are described well by the Standard Model (SM) of particle physics, but new physics must be responsible for the dark matter, the matter-antimatter asymmetry, etc [1]. It is quite possible that a more complete theory with additional gauge interactions may provide solutions to those anomalies. This motivates experimental searches for the non-SM gauge bosons, named X in this work, which mediate such extended interactions (new forces). In general, the X can be either a neutral massive scalar/psedoscalr, a Z 0 -like, γ-like (dark photon), or half spin particle.
Here, we focus on the search of new force carrier X at BESIII detector. The BESIII experiment works in the C.M.S. energy region of 2 ∼ 4.6 GeV and has accumulated 1.3 billion J/ψ's and 0.5 billion ψ(3686)'s, which provides ideal samples for the new force carrier search in the mass region of MeV to several GeV [20]. The latest result is the dark photon (U) search in J/ψ → η ′ U followed by U → e + e − decay. The mass region of 0.1 ∼ 2.1 GeV was explored but no significant signal observed [21]. The initial state radiation processes e + e − → e + e − γ ISR and e + e − → µ + µ − γ ISR were scanned in the mass region 1.5 up to 3.4 GeV, yet no enhancement found in the invariant-mass spectrum of leptonic pairs [22]. Using the abundant J/ψ and ψ ′ samples, the decay chains of J/ψ → A 0 γ, A 0 → µ + µ − [8] and ψ ′ → J/ψπ + π − , J/ψ → A 0 γ, A 0 → µ + µ − [9] are employed to searches for the light CP-odd Higgs-like particle A 0 .
Theoretically, when considering the interactions between the new force mediator X and SM particles, extra Lagrangian needs to be added to the SM. For the dark photon, the coupling arises from the gauge-invariant "kinetic mixing" of the new Abelian gauge group U(1) X with the SM hypercharge group U(1) Y [23,24]. For a Z 0 -like X boson, the theory with axial-vector couplings can also be UV-completed consistent with SM gauge invariance [25,26]. And CP-odd pseudoscalar Higgs bosons are suggested in the next-to-minimal super-symmetric standard model [27], where the mass of the lightest CP-odd Higgs boson may be less than J/ψ. In phenomenological study, various possible phenomenons with the new force carrier X were investigated. One representative example, a fifth force mediated by a protophobic 17 MeV boson was suggested to explain the recent 8 Be * anomaly [28][29][30], which might also be a solution to the muon's anomalous magnetic moment [29,30], NuTeV anomaly [31] or 511 keV line [32]. In addition, various literatures also discuss the possibility of the search for the new force carrier X in current and future experiments, see Refs. [33][34][35][36][37][38] for instance.
In this work, we study the production of the new force mediator X boson associated with a photon in both electron-positron collision and J/ψ decay at BESIII, taking account the different theoretical hypotheses on the nature of X. The exclusion limits on the reduced coupling strength of X boson to SM particles as functions of the X boson mass in BESIII experiment condition will be presented. To be noted, Li and Luo [35] also discussed the search of the dark photon and a Higgs-like boson at BESIII, whereas in a different method from this work.
As a mediator, the new force carrier X is usually taken to be a spin-0 or spin-1 boson.
In this section, we study the production and decay properties of X boson in electronpositron collision for both spin-0 and spin-1 hypotheses, in particular, the exclusion limit on the reduced coupling strength parameter as a function of X boson mass under the BESIII experiment condition will be discussed. It needs to be clarify that several formulas presented in Z 0 -like X case are overlapped with those in our previous work [39], but the focuses here are different.

A. Spin-1 Hypothesis
As a general case, new Lagrangian of the spin-1 X boson can be formulated as where e is the electron charge and ǫ v/a denote the reduced coupling strength of new boson X to vector/axial-vector currents, which implies the X boson can either be a γ-like or a Z 0 -like particle. In Eq. (1), we simply assume the reduced coupling strengths of the new particle to leptons and quarks are equal.
FIG. 1: The Born level Feynman diagrams of e + e − → X + γ process, where the new particle X can be a massive neutral spin-0 or spin-1 particle.
We consider the production of the new boson X associated with a SM photon in e + e − collision, whose Born level Feynman diagrams are displayed in Fig. 1. And the differential cross section of the process e + e − → X + γ can be readily obtained, with Here θ is the emitting angle of photon with respect to the e + e − beam axis, √ s is the C.M.S energy, m X/e are the masses of the X/electrons, and α = 1/137 is the fine structure constant.
With Eq. (2), one can easily obtain the differential distribution of the cross section with respect to cosθ and the total cross section as a function of C.M.S energy √ s, which have been presented in Fig. 2. Here we consider four m X inputs, and the reduced coupling strength parameters ǫ v = ǫ a = 10 −3 . Since the cross section is proportional to the squared reduced coupling strength parameters ǫ 2 v/a , one can easily estimate the results when adopting other ǫ v/a inputs. From Fig. 2, we find that both the differential distribution dσ/dcosθ and the total cross section are not sensitive to the new boson mass MeV, 1 GeV and 2 GeV respectively. And ǫ v = ǫ a = 10 −3 is adopted.
Running at √ s = 3.7 GeV, the luminosity of BESIII can reach 10 33 cm −2 s −1 ≃ 10 4 pb −1 year −1 . Then we can estimate the events of the Z 0 -like X boson produced per year as a function of its mass m X in e + e − → X + γ process, as is presented in Fig. 3. Here we have taken the 93% solid coverage of BESIII detector into consideration. It is found that the values grow slowly in small m X region. And events of the new boson X at m X ≈ 2.5 GeV are about two times of that below 1 GeV.
Experimentally, we would reconstruct the new boson X with its decay products.
Since its mass m X may range from tens of MeV to several GeV at BESIII, the decay products can be complicated. Below the 2π threshold (∼ 270 MeV), the new boson X can decay into e + e − , µ + µ − , photons, νν or light dark matters. While heavier X boson may have various hadronic decay products. In this manuscript, we use the electronpositron pairs to fully reconstruct the new boson X events, i.e. X → e + e − . And the decay width is Below the 2µ threshold (∼ 210 MeV), since the 2γ/3γ decay modes are highly suppressed in the Z 0 -like X hypothesis, we can assume reasonably the X → e + e − decay is saturable.
In the experimental frame of e + e − → X + γ process, the velocity of X boson is v = It is meaningful to measure the decay length of X boson since it can help to determine the coupling strength ǫ and help to identify the signals over the background.
For the γ-like X boson (dark photon) case, one can obtain the cross section or decay width by setting ǫ a = 0 in the relevant formulas above. We find that numerical values of the differential distribution dσ/d cos θ and total cross section or events as functions of √ s or m X for the γ-like X boson are half of those displayed in Fig Of the concerned process e + e − → e + e − γ, the propagator can be either a X boson (signals) or a virtual photon (background), whose Feynman diagrams are presented in  Under the BESIII experiment conditions, we evaluate the ratio σ sig /σ bac as a function of the X boson mass m X , where σ sig /σ bac are the cross sections of signal/bacground processes e + e − X/γ * − −− → e + e − γ respectively, as is presented in Fig. 5. Here the contribution of Feynman diagrams Group (II) is plotted separately in comparison with that of the total (I + II). The cross section of the background σ bac = 31 nb according to our estimate. For the γ-like X boson case, we evaluate the ratio σ Z 0 −likeX /σ γ−likeX as a function of m X , which is also displayed in Fig. 5. It is clear that the contribution of Group (II) dominates the cross section, especially when Γ X is small and m X is large.
And the ratio σ Z 0 −likeX /σ γ−likeX = 4 for the Group (II) implies that the axial-vector current and the vector one are of the same importance. It is worth noting that here we adopt the experimental selection conditions at BESIII, i.e. the photon selection condition is | cos α| < 0.8 with the energy E γ > 25 MeV for the barrel 1 , while good charged tracks are constrained in the region of | cos β| < 0.93 [40], with α/β being the polar angles of final particles with respect to the e + e − beam axis.  And we also find that the exclusion limits increase with the decrease of the Γ X . This can be attributed to the dominated Feynman diagrams Group (II), since the smaller Γ X in the Breit-Wigner propagator of diagrams (II) will result in bigger cross section at the X resonance point of M ee = m X . In fact, because of the factor ǫ 2 v/a , the X decay width Γ X maight be mush smaller than the values we adopted above, especially for light X boson. One may refer to the X(16.7) boson case discussed in our previous work [39]. In this subsection, we will extend our analysis and search for the neutral spin-0 X particle in the e + e − → e + e − γ process. We define the new Yukawa interaction Lagrangian for the scalar(S)/pseudoscalar(PS) X respectively, Here e is the electron charge, η is the reduced Yukawa coupling strength of scalar X to fermions, and ξ stands for that of pseudoscalar X to fermions. Here we also set the Yukawa coupling strength of the X boson to leptons and quarks to be equal, since the cross sections are always proportional to them.
For the scalar/pseudoscalar X boson, we also consider its production in the process of e + e − → X + γ. Then we obtain the differential cross section for scalar/pseudoscalar X with respect to cosθ respectively, with where variables have the same meanings as those in Eq. (2). As for the decay of scalar/pseudoscalar X boson, it becomes complicated since the loop induced decay mode X → 2γ is possible. So we will simply set its total decay width Γ X as an input parameter in the following discussion.
It's worth noting that, in the coming numerical analysis, we find that all the curves in the scalar X case are overlapped with those in the pseudoscalar X case, including the curve of exclusion limits on the reduced coupling parameter versus the mass m X . So we only discuss the results of scalar X case as a fine example in this subsection.
With the help of Eq. (5), we can evaluate the differential distribution of the cross section with respect to cosθ and the total cross section as a function of C.M.S energy √ s for scalar X boson, which are presented in Fig. 7. For the detection of scalar X at BESIII, we also use the electron-positron pairs to fully reconstruct the signals, and the corresponding Feynman diagrams of e + e − X − → e + e − γ are the same as those in Fig. 4. Then we evaluate the ratio σ sig /σ bac as a function of the X boson mass m X , where σ sig /σ bac are the cross sections of signal/bacground processes, as is presented in Fig. 9. Where the contribution of Feynman diagrams Group (II) is plotted separately in comparison with the total (I + II) case. In Fig. 9, we also present the exclusion limit on the reduced coupling strength parameter η versus the boson mass m X for the (I + II) case. Note that the BESIII experiment selection conditions are also adopted. One can find that the upper limit on the parameter η lies in between 10 −3 and through e + e − → X + γ process at √ s=3.7 GeV. η = 10 −4 is adopted.
10 −2 depending on m X for Γ X = 0.1 ∼ 10 MeV, which is similar to the result of Z 0 -like X boson case.
In fact, the BESIII experiment once searched for the dark photon in the initial state radiation reactions e + e − → e + e − γ ISR and e + e − → µ + µ − γ ISR using a data set of 2.93 fb −1 at √ s = 3.77 GeV [22]. But no enhancement is observed in the mass region of 1.5 This, in the other way around, indicates that the decay width Γ X of X boson should be constrained below 0.1 MeV.

III. X BOSON IN J/ψ DECAY
As is known that BESIII has collected the largest J/ψ data in the world, more than (1310.6 ± 7.0) × 10 6 [41]. So in this section we will search for the new boson X in J/ψ → e + e − γ decay, and give the exclusion limit on the reduced coupling strength parameter versus the X boson mass. Note that partial formulas presented in Z 0 -like X case overlap with those in our previous work [39].

A. Axial-vector Hypothesis
Due to the fact that J/ψ cannot decay into a γ-like X boson and a photon, we will consider the Z 0 -like X hypothesis in this subsection. Then adopting the "vector minus axial-vector" interaction Lagrangian in Eq. (1), one can readily obtain the decay width of J/ψ → X + γ process, where m c = 1.5 GeV and the squared wave function at the origin Ψ 2 = m 2 c Γ(J/ψ→e + e − ) 4πe 2 c α 2 (1−8αs/(3π)) [42], with e c = 2/3, α s = 0.23 and Γ(J/ψ → e + e − ) = 5.55 × 10 −6 GeV [1]. Obviously, only the axial-vector current survives here. Given the total decay width of J/ψ as 92.9 keV and the collected 1.3×10 9 J/ψ events, we can estimate the events of this new Z 0 -like X boson as (6.4, 6.7, 15, 65) × 10 7 ǫ 2 a for m X = (0.02, 0.2, 1, 2) GeV respectively. Next, we consider identifying this Z 0 -like X signals in the J/ψ → e + e − γ process, and both the X propagated Feynman diagrams (signals) and the virtual photon propagated ones (background) are presented in Fig. 10. One may notice that the background process has only two diagrams of Fig. 10 (a, b), while the signal process has two extras of Fig. 10 (c, d). Here, we also adopt the Breit-Wigner form for the propagators in the calculation of Fig. 10 (c, d). Then we can calculate the differential decay width of these signal/background processes with respect to the Dalitz invariants s 1 and s 2 , which have been sorted in Appendix. In Fig. 11, we present the ratio Γ sig /Γ bac as a function of the X boson mass m X , and Γ sig /Γ bac are the decay widths of signal/bacground processes J/ψ respectively. Here ǫ v = ǫ a = ǫ is assumed, the invariant-mass of e + e − pairs M ee ( √ s 1 ) is constrained in 20 MeV< M ee < 2.98 GeV, and the decay width Γ bac = 6.6 × 10 −3 Γ J/ψ , which is consistent with the PDG data [1]. The solid, dashed and dot-dashed lines are for total decay widths of X boson Γ X = 0.1, 1 and 10 MeV respectively. Here we find the X resonant Feynman diagrams of Fig. 10 (c, d) dominate the decay width. The regions of the X boson parameter space (ǫ vs m X ) are presented in Fig. 12. It is found that, with the 1.3 × 10 9 J/ψ events, the upper limit on the parameter ǫ is placed in between 10 −3 and 10 −2 depending on m X and Γ X .
The Feynman diagrams of J/ψ → e + e − γ process mediated by scalar/pseudoscalar X boson are the same as those for Z 0 -like X case. Notice that only Fig. 10 (c, d) contribute because of the conservation of orbital angular momentum. And we also evaluate the differential decay widths dΓ/(ds 1 ds 2 ) for both scalar/pseudoscalar cases, with , .
With these expressions, one can derive the ratios Γ sig /Γ bac as functions of the X boson mass for scalar/pseudoscalar X boson respectively, as displayed in Fig. 13. For the exclusion limits on the reduced coupling strength parameters as functions of m X at BESIII detector, the parameter spaces (η/ξ vs m X ) for scalar/pseudoscalar X boson are presented in Fig. 14   According to our analysis, at BESIII the exclusion limits on the reduced coupling strengths lie in-between 10 −3 to 10 −2 depending on m X , suppose the decay width of X boson Γ X = 0.1 ∼ 10 MeV. Note that the upper limit may be increased with the decrease of Γ X .
We find that the Z 0 -like events produced in e + e − → Xγ are more than those events with X being scalar/pseudoscalar or dark photon, and can reach (10 9 ∼ 10 10 ) × ǫ 2 per year depending on m X . This suggests that the Z 0 -like boson X signal might be found in the product chain e + e − → Xγ, X → l + l − (l = e, µ), or excluded, in the present run of BESIII, especially when its mass below the 2π threshold where the Γ X tends to be small. While for X boson indirect production in J/ψ decay, analysis shows that a much big database of J/ψ is required.

Background process
We firstly introduce some notations, s 1 = (k 1 +k 2 ) 2 , s 2 = (k 2 +k 3 ) 2 , Ψ 2 is the squared wave function at the origin of J/ψ and Γ X is the total decay width of new boson X.